Artin-Wedderburn theorem and representation theory

Question

How to apply the Artin-Weddernburn theorem to representation theory? The Artin-Weddeburn theorem says that if $R$ is finite dimensional $k$ algebra( where $k$ is an algeraically closed field), then $R=M_{n_1}(k)\bigoplus\dots\bigoplus M_{n_r}(k)$ for some $r$ and $n_i$. OTOH, we know from Maschke's theorem that if $G$ is a finite group and char$(k)\nmid|G|$, $k[G]$ is semisimple. So we can write $k[G]=M_{n_1}(k)\bigoplus\dots\bigoplus M_{n_r}(k)$. Then can we get any information about the $k$ linear representation of $G$ from this decomposition? For example, can we conclude that the dimensions of the irreducible representations of $G$ are $n_1,\dots,n_r$ and the number of conjugacy classes of $G$ is $r$? Can we apply a-w theorem to representation theory if $k$ is not algebraically closed?